### Abstract

Let G be a locally compact group with left Haar measure m_{G} on the Borel sets IB(G) (generated by open subsets) and write |E|=m_{G}(E). Consider the following geometric conditions on the group G. (FC If e{open}>0 and compact set K⊂G are given, there is a compact set U with 0<|U|<∞ and |x U ΔU|/|U|<e{open} for all xεK. (A) If e{open}>0 and compact set K⊂G, which includes the unit, are given there is a compact set U with 0<|U|<∞ and |K U ΔU|/|U|<e{open}. Here A ΔB=(A/B){smile}(B/A) is the symmetric difference set; by regularity of m_{G} it makes no difference if we allow U to be a Borel set. It is well known that (A)⇒(FC) and it is known that validity of these conditions is intimately connected with "amenability" of G: the existence of a left invariant mean on the space CB(G) of all continuous bounded functions. We show, for arbitrary locally compact groups G, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact group G satisfies. (C) For at least one set K, with int(K)≠Ø and {Mathematical expression} compact, there is an indexed family {x_{α}:αεJ}⊂G such that {Kx_{α}} is a covering for G whose covering index at each point g (the number of αεJ with gεKx_{α}) is uniformly bounded throughout G.

Original language | English (US) |
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Pages (from-to) | 370-384 |

Number of pages | 15 |

Journal | Mathematische Zeitschrift |

Volume | 102 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 1967 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*102*(5), 370-384. https://doi.org/10.1007/BF01111075